The standardised formula worked step by step on a fictional SME pool, and why a standardised bank has to sell a much thicker tranche than an IRB bank to get the same relief.
Under SEC-SA, compute KA = (1 − W) × KSA + 0.5 × W, set a = −1 / (p × KA) with p = 1, and the tranche weight is 12.5 times the supervisory formula KSSFA, floored at 15 per cent. On an illustrative 2,000m SME pool weighted at 85 per cent, a senior attaching at 12 per cent weighs 44.96 per cent and frees 122.7m of capital at a 13.50 per cent CET1 target. It reaches the 15 per cent floor only at a 20.12 per cent attachment.
Worked in full in Significant Risk Transfer by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A bank without IRB permission for the securitised loans cannot compute KIRB, so the hierarchy sends it to the standardised approach of Article 261 of the CRR. The structure of the formula is the same as SEC-IRBA: a capital charge for the pool, a tranche that either sits below it and is weighted at 1250 per cent or above it and is charged on a decaying exponential. Two things differ. The anchor is KA, the standardised capital of the pool adjusted for delinquent loans, and the p factor is fixed at 1 for a non-STS deal, against a computed p that is usually much lower under SEC-IRBA. That second difference is what makes standardised SRT expensive.
| Input | Value |
|---|---|
| Pool exposure | 2,000 |
| Average standardised risk weight | 85% |
| KSA = 8% × average risk weight | 6.80% |
| Delinquent share W at closing | 0% |
| p factor, non-STS / STS | 1 / 0.5 |
| Junior tranche sold, cash-collateralised | 0 to 12%, 240 |
| Senior retained | 12 to 100%, 1,760 |
| Risk weight floor, non-STS / STS senior | 15% / 10% |
| CET1 target | 13.50% |
The junior is sold through credit-linked notes whose proceeds sit in cash with the bank, so it carries no risk weight. Retention is left out for clarity; how it is weighted is in vertical slice or random selection.
Step 1: KA. KA = (1 − W) × KSA + 0.5 × W = 1 × 6.80% + 0 = 6.80%
Step 2: the exponent. a = −1 / (p × KA) = −1 / (1 × 6.80%) = −14.706
Step 3: the tranche measured from KA. u = D − KA = 100% − 6.80% = 93.200%; l = max(A − KA, 0) = 12% − 6.80% = 5.200%
Step 4: KSSFA. KSSFA = (ea·u − ea·l) / (a × (u − l))
a·u = −13.706, so ea·u = 0.0000011; a·l = −0.7647, so ea·l = 0.4655; a × (u − l) = −12.941
KSSFA = (0.0000011 − 0.4655) / −12.941 = 3.597%
Step 5: the weight. RW = max(12.5 × 3.597%, 15%) = 44.96%
Excel, with KA in B1 and A in B2: =MAX(12.5*(EXP(-1/B1*(1-B1))-EXP(-1/B1*MAX(B2-B1,0)))/(-1/B1*(1-B1-MAX(B2-B1,0))),0.15)
If the tranche sits entirely below KA the weight is 1250 per cent. If it straddles KA, the weight is the 1250 per cent on the slice below KA and 12.5 × KSSFA on the slice above, averaged by thickness.
| Line | RWA | Capital at 13.50% |
|---|---|---|
| Pool before: 2,000 × 85% | 1,700.0 | 229.5 |
| Senior after: 1,760 × 44.96% | 791.3 | 106.8 |
| Released | 908.7 | 122.7 |
Selling 240m of protection removes 53.5 per cent of the pool's RWA. That is respectable, but it took a 12 per cent tranche, almost twice KA, to get there. On an IRB pool the senior usually reaches its floor within a few points of KIRB, as the SEC-IRBA worked example shows. With p = 1, the senior here would need to attach at 20.12 per cent to reach 15 per cent.
| Attachment | Protection sold | Senior RW, non-STS | Capital released | Per 100 sold | Senior RW, STS | Capital released, STS |
|---|---|---|---|---|---|---|
| 8% | 160 | 77.44% | 37.1 | 23.2 | 32.46% | 148.9 |
| 9% | 180 | 67.59% | 63.4 | 35.2 | 24.45% | 169.4 |
| 10% | 200 | 58.99% | 86.1 | 43.1 | 18.42% | 184.7 |
| 12% | 240 | 44.96% | 122.7 | 51.1 | 10.46% | 204.6 |
| 14% | 280 | 34.28% | 149.9 | 53.5 | 10.00% | 206.3 |
| 16% | 320 | 26.16% | 170.2 | 53.2 | 10.00% | 206.8 |
| 18% | 360 | 19.97% | 185.3 | 51.5 | 10.00% | 207.4 |
| 20% | 400 | 15.25% | 196.6 | 49.1 | 10.00% | 207.9 |
Two readings matter. Capital released per 100 of protection peaks around 14 to 16 per cent and then falls, because each extra point of thickness buys less: 59.2m of capital from 9 to 12 per cent, 38.1m from 12 to 15, 24.5m from 15 to 18. And STS changes the shape completely. With p = 0.5 the senior hits its 10 per cent floor at a 12.16 per cent attachment, so an STS deal gets 204.6m from the same 240m tranche. For a standardised bank the label is worth more than for an IRB bank; the IRB side of that comparison is in how much capital STS status saves.
KSA is set by the risk weights, but W moves with the pool. Eligibility criteria usually keep delinquent loans out at closing, so W starts near zero. It does not stay there, and SEC-SA has no lag: the weight is recomputed on each reporting date with the delinquencies the pool has at that time.
| W | KA | RW at 12% | RW at 9% |
|---|---|---|---|
| 0% | 6.80% | 44.96% | 67.59% |
| 1% | 7.23% | 53.13% | 77.80% |
| 2% | 7.66% | 61.83% | 88.43% |
| 5% | 8.96% | 90.65% | 122.52% |
With 2 per cent of the pool delinquent, KA = 0.98 × 6.80% + 0.02 × 50% = 7.664%, and the senior's weight rises from 44.96 to 61.83 per cent: 40.1m more capital on the same structure. At 5 per cent, a senior attaching at 9 per cent weighs more than the 85 per cent the whole pool weighed at closing, and that is before the delinquent loans push KSA itself up. A bank that models the deal once, at closing, overstates the relief it can count on through a downturn, which is exactly when it needs it.
The weights here follow the formula of Article 261 and are illustrative. Whether the relief is recognised at all depends on the significant risk transfer tests and the supervisor, not on the formula.
The book works the same mechanics on an IRB pool, from KIRB through SEC-IRBA to the bank's economics, and the free workbooks for this case include a SEC-IRBA calculator for any tranche you enter.
SEC-IRBA anchors the formula on KIRB, the IRB capital of the pool, and computes the p factor from the pool and tranche. SEC-SA anchors it on KA, the standardised capital adjusted for delinquent loans, and fixes p at 1 for a non-STS deal. The higher p makes capital fall away more slowly above KA, so on the illustrative pool the senior reaches the 15% floor only at a 20.12% attachment.
W is the share of the pool that is delinquent: 90 days past due, in default, in foreclosure or subject to bankruptcy proceedings. It enters KA with a 50% weight. On the illustrative pool, delinquencies of 2% lift KA from 6.80% to 7.664% and the senior's weight at a 12% attachment from 44.96% to 61.83%, 40.1m more capital at 13.50%.
Enough that the retained senior weighs little against the pool's own standardised weight. On an 85% weighted pool with no delinquencies, the senior weighs 67.59% at a 9% attachment, 44.96% at 12% and 19.97% at 18%. Each extra point of thickness releases less: 59.2m of capital from 9 to 12%, 24.5m from 15 to 18%.
This article is one calculation from Significant Risk Transfer. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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